Only a refusal by the leaders of the 27 remaining EU states to grant the UK an extension at a Brussels summit next week could now preserve the totemic date of March 29 as Brexit Day.

Mrs May has made clear that she will press her Agreement to a third “meaningful vote” in the Commons by March 20 in the hope of securing the support of MPs who rejected it by 230 votes in January and 149 earlier this week.

If she succeeds, she will go to Brussels next Thursday to request a short delay to a date no later than June 30, to give herself time to pass legislative changes necessarily for a smooth and orderly Brexit.

But if her deal is rejected for a third time, she believes any extension would have to be far longer and would involve the UK taking part in European Parliament elections in May.

Cabinet Office minister David Lidington told MPs that in this case, the Government would stage two weeks of debate following the March 21-22 summit for the Commons to try to establish a majority around a different plan.

Ministers met for a reportedly testy political meeting of Cabinet ahead of the votes, at which Mrs May was said to have berated four senior colleagues who defied the Tory whip to abstain in the no-deal vote.

European Council president Donald Tusk has indicated that the EU may be ready to offer a lengthy extension to negotiations if the UK wants to “rethink its Brexit strategy and build consensus around it”.

But any extension must be approved unanimously by the EU27, and Austrian foreign minister Karin Kneissl said there could be “some problem” in obtaining this if it took Brexit beyond the date of elections.

Taoiseach Leo Varadkar said: "I welcome the House of Commons vote for an extension as it reduces the likelihood of a cliff edge, no-deal Brexit on March 29th. But we now need to hear from London about what purpose an extension would serve and how long it would last.

"There seem to be two emerging options: ratification of the withdrawal agreement followed by a short extension into the summer, or a much longer extension that would give the UK time and space to decide what they want to do, including considering options that had been taken off the table like participation in the customs union and single market.

"I think we need to be open to any request they make, listen attentively and be generous in our response.

"This matter will be now discussed further at next week’s European Council meeting and hopefully we will have more clarity from London in the meantime about their intentions.

"Unanimity by the 27 will be required for any extension."

Tánaiste Simon Coveney told RTE radio: “If you have a long extension of, say, 21 months to the end of 2020 – whatever the period would be – then Britain has a legal entitlement to have representation in the European Parliament” and so must take part in the elections.

Earlier, MPs decisively rejected an attempt by the Independent Group to secure a second referendum on Brexit by 334 votes to 85.

Mr Bercow sparked fury among Brexiteers by selecting TIG MP Sarah Wollaston’s referendum proposal for debate, while blocking another amendment which sought to rule out a second public vote.

And by the far narrower margin of 314-312, they voted down a cross-party bid for Parliament to seize control of the Brexit process.

The cross-party amendment, tabled by MPs including Labour’s Hilary Benn and Yvette Cooper and Tory Sir Oliver Letwin, would have forced a set of “indicative votes” to determine the preferred Brexit outcome of the House of Commons.

A Labour amendment demanding an extension to Article 50 withdrawal negotiations to provide time to “find a majority for a different approach” was also defeated.The votes came as US President Donald Trump said Brexit was ripping Britain apart and warned that another referendum would be “unfair”.

Speaking during Taoiseach Leo Varadkar’s visit to the White House, Mr Trump said: “I’m surprised at how badly it’s all gone from the standpoint of a negotiation. I gave the Prime Minister my ideas on how to negotiate it.

“She didn’t listen to that and that’s fine – she’s got to do what she’s got to do.

“I think it could have been negotiated in a different manner, frankly. I hate to see everything being ripped apart now.”

During my consultations ahead of #EUCO, I will appeal to the EU27 to be open to a long extension if the UK finds it necessary to rethink its #Brexit strategy and build consensus around it.

A spokesman for the People’s Vote campaign said they regarded Dr Wollaston’s decision to press the referendum case to a vote as premature.

“We do not think today is the right time to test the will of the House on the case for a new public vote,” said the spokesman. “Instead, this is the time for Parliament to declare it wants an extension of Article 50 so that, after two-and-a-half years of vexed negotiations, our political leaders can finally decide on what Brexit means.”

Labour whipped its MPs to abstain on the referendum vote, but 24 voted in favour – not including Brighton’s Lloyd Russell-Moyle, who went through both lobbies to cancel his own vote out.

TIG Brexit spokeswoman Anna Soubry said: “This is a betrayal of Labour Party members and voters, Labour MPs, Labour’s conference policy and, most importantly, the British public. The Labour Party leadership are determined to deliver Brexit, which would harm our country.”

Labour revealed that leader Jeremy Corbyn and senior aides have met with backbenchers Peter Kyle and Phil Wilson, who are promoting a plan to accept Mrs May’s deal on the condition that it is subject to a second referendum.